Tag-Trigger-Consolidation: A Model of Early and Late Long ... · protein kinase Mf (PKMf) [11,14]. The stabilization and maintenance of potentiated synapses poses a number of theoretical

Tag-Trigger-Consolidation: A Model of Early and LateLong-Term-Potentiation and DepressionClaudia Clopath., Lorric Ziegler., Eleni Vasilaki, Lars Busing¤, Wulfram Gerstner*

Laboratory of Computational Neuroscience, Brain-Mind Institute and School of Computer and Communication Sciences, Ecole Polytechnique Federale de Lausanne,

Lausanne, Switzerland

Abstract

Changes in synaptic efficacies need to be long-lasting in order to serve as a substrate for memory. Experimentally, synapticplasticity exhibits phases covering the induction of long-term potentiation and depression (LTP/LTD) during the early phaseof synaptic plasticity, the setting of synaptic tags, a trigger process for protein synthesis, and a slow transition leading tosynaptic consolidation during the late phase of synaptic plasticity. We present a mathematical model that describes thesedifferent phases of synaptic plasticity. The model explains a large body of experimental data on synaptic tagging andcapture, cross-tagging, and the late phases of LTP and LTD. Moreover, the model accounts for the dependence of LTP andLTD induction on voltage and presynaptic stimulation frequency. The stabilization of potentiated synapses during thetransition from early to late LTP occurs by protein synthesis dynamics that are shared by groups of synapses. The functionalconsequence of this shared process is that previously stabilized patterns of strong or weak synapses onto the samepostsynaptic neuron are well protected against later changes induced by LTP/LTD protocols at individual synapses.

Citation: Clopath C, Ziegler L, Vasilaki E, Busing L, Gerstner W (2008) Tag-Trigger-Consolidation: A Model of Early and Late Long-Term-Potentiation andDepression. PLoS Comput Biol 4(12): e1000248. doi:10.1371/journal.pcbi.1000248

Editor: Lyle J. Graham, UFR Biomedicale de l’Universite Rene Descartes, France

Received August 18, 2008; Accepted November 10, 2008; Published December 26, 2008

Copyright: � 2008 Clopath et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permitsunrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: This work was partially supported by the European community via th FACETS project. CC was supported by the Swiss National Science Foundation. Thesponsors did not influence the design or analysis of the study.

Competing Interests: The authors have declared that no competing interests exist.

* E-mail: [email protected]

¤ Current address: Institut fur Grundlagen der Informationsverarbeitung, TU Graz, Graz, Austria

. These authors contributed equally to this work.

Introduction

Changes in the connection strength between neurons in

response to appropriate stimulation are thought to be the

physiological basis for learning and memory formation [1,2]. A

minimal requirement for proper memory function is that these

changes, once they are induced, persist for a long time. For several

decades, experimentalists have therefore focused on Long-Term

Potentiation (LTP) and Long-Term Depression (LTD) of synapses

in hippocampus [3,4] and cortical areas [5,6]. LTP can be induced

at groups of synapses by strong ‘tetanic’ high-frequency stimula-

tion of the presynaptic pathway [3] while stimulation at lower

frequency leads to LTD Dudek92. Both LTP and LTD can also be

induced at a single synapse or a small number of synaptic contacts

if presynaptic activity is paired with either a depolarization of the

postsynaptic membrane [5,7] or tightly timed postsynaptic spikes

[8,9].

While the induction protocol for LTP and LTD is often as short

as a few seconds, the changes in synaptic efficacy persist for much

longer [9]. In typical slice experiments on LTP [and similarly for

LTD or Spike-Timing Dependent Plasticity (STDP)] the persis-

tence of the change is monitored for 30 minutes to 1 hour.

Accumulating evidence suggests, however, that after this early

phase of LTP (E-LTP) different biochemical processes set in that

are necessary for the further maintenance of potentiated synapses

during the late phase of LTP (L-LTP) [10,11]. For an

understanding of the transition from early to late LTP, the

concept of ‘synaptic tagging and capture’ has become influential

[12,13]. During induction of the early phase of LTP, each

potentiated synapse sets a tag that marks that it has received a

specific afferent signal. A candidate molecule, involved in the tag

signaling LTP induction in apical dendrites of hippocampal

neurons, is the calcium-calmodulin dependent kinase II (CaMKII)

[13]. Newly synthesized plasticity-related proteins are ‘captured’

by the tagged synapse and transform E-LTP into L-LTP that can

be maintained over hours or days. A candidate protein involved in

the maintenance of potentiated hippocampal synapses is the

protein kinase Mf (PKMf) [11,14].

The stabilization and maintenance of potentiated synapses

poses a number of theoretical challenges. First, on the level of

single synapses we must require synaptic strength to remain stable,

despite the fact that AMPA channels in the postsynaptic

membrane are continuously exchanged and recycled [15–17].

Thus the synapse is not ‘frozen’ but part of a dynamic loop.

Second, on the level of neuronal representation in cortical areas,

one finds representations of input features that are stable but at the

same time sufficiently plastic to adjust to new situations [18]. In the

theoretical community, this paradox has been termed the stability-

plasticity dilemma in unsupervised learning [19]. Third, humans

keep the ability to memorize events during adulthood, but can also

remember earlier episodes years back. However, continued

learning of new patterns in theoretical models of associative

memory networks forces the erasure or ‘overwriting’ of old ones,

the so-called palimpsest property [20,21]. In the context of

PLoS Computational Biology | www.ploscompbiol.org 1 December 2008 | Volume 4 | Issue 12 | e1000248

continued learning, theoretical arguments show that synaptic

plasticity on multiple time scales cannot prevent, but at most delay

the erasure of memories in the presence of ongoing synaptic

activity [22]. This suggests that additional mechanisms are

necessary to further protect existing memories and ‘gate’ the

learning of new ones.

Despite these challenges for the long-term stability of synapses,

most classical models of synaptic plasticity focus on the induction

and early phase of LTP or LTD and completely ignore the

question of maintenance. Traditional models of associative

memories separate the learning phase from the retrieval phase

[23] and the same holds for standard models of STDP [24–26].

Detailed biophysical models of LTP and LTD describe calcium

dynamics and Calcium/Calmodulin-Dependent Protein Kinase II

(CaMKII) phosphorylation during the induction and early phase

of LTP [27–29]. While these models show that switches built of

CaMKII proteins can be stable for years, they do not address

aspects of tagging leading to heterosynaptic interaction during L-

LTP and L-LTD. Moreover, while CaMKII phosphorylation is

necessary for induction of LTP and mediate tags in the apical

dendrites of hippocampal CA1 neurons [30], it is less clear

whether it is necessary for its maintenance [31]. On the other hand

protein kinase Mf is essential for maintenance of some synapse

types [11,13,14] but the same molecule is potentially relevant for

induction in others [30].

We wondered whether a simple model that connects the process

of LTP induction with that of maintenance would account for

experimental results on tagging and ‘cross-tagging’ [11–13,32]

without specific assumptions about the (partially unknown)

molecular pathways involved in the maintenance process. If so,

the model should allow us to discuss functional consequences that

are generic to the tagging hypothesis independent of the details of

a biophysical implementation in the cell. Even though we believe

that the model principles are more general, we focus on synapses

from the Schaffer-Collaterals onto the CA1 neurons in hippo-

campus as an experimentally well-studied reference system for

synaptic plasticity. Since typical tagging experiments involve the

extracellular stimulation of one or several groups of synapses (rather

than single synapses), our model of early and late LTP/LTD is

developed in the context of a neuron model with hundreds of

synapses. The application of the principles of synaptic consolida-

tion to experiments inducing E-LTP/E-LTD at single synapses is

considered in the discussion section.

Results

We study a model with a large number of synapses i onto a

single postsynaptic neuron. To be specific, we think of a pyramidal

neuron in the CA1 area of hippocampus. Our model combines

features of traditional models for the induction of potentiation [24–

26,33–36] with a simple description of tagging and synthesis of

plasticity related proteins that finally lead to the maintenance of the

induced changes. The section is organized as follows: We first

introduce the essential components of the model step by step

(‘Constructing the Model’). We then test the performance of the

model with a set of stimuli typically used to induce long-term

changes of synapses (‘Testing the Model’).

Constructing the ModelOur model contains three elements, Figure 1. The first one sets

the tag during the induction of E-LTP or E-LTD. A tag is

indicated by a value h = 1 for LTP or l = 1 for LTD. In the absence

of tags we have h = l = 0. The second one describes the process that

triggers the synthesis of plasticity related proteins. The final

component describes the up-regulation of a maintenance-related

process from a low value (z = 0) to a high value (z<1). The

dynamics of this component is intrinsically bistable and leads to a

consolidation of the previously induced change at the labeled

synapses upon interaction with the protein p (‘protein capture’).

The total change Dw of the synaptic strength reported in

experiments contains contributions [13] of the early components

l and h as well as the late component z. Since the model describes a

sequence of three steps ‘Tag-Trigger-Consolidation’ we call it in

the following the TagTriC-Model (Figure 1).

Tag and Induction of LTP/LTDResults from minimal stimulation protocols which putatively

activate only a single synapse suggest that the induction of LTP is a

switch-like process [7,37]. We therefore model individual synapses

as discrete quantities that can switch, during the induction of LTP,

from an initial ‘non-tagged state’ (N) to a ‘high state’ (H) with a

transition rate rH that depends on the induction protocol.

Similarly, induction of LTD moves the synapse from the initial

non-tagged state (N) to a ‘low state’ (L) at a rate rL. If synapse i is

in the high state, the synaptic variable hi is equal to one. If it is in

the low state, another local variable li is set to one. These local

variables hi and li do not only control the weight of the synapse

during E-LTP and E-LTD, but also serve as ‘tags’ for up- or

down-regulation of the synapse. Tags reset to zero stochastically

with a rate kh and kl, respectively. If both tags are zero, the synapse

is in the non-tagged state N. Since the synapse is either up-

regulated OR down-regulated, at most one of the tags can be non-

zero (Figure 1A).

The stochastic transitions from the initial state N with hi = 0 and

li = 0 to the down-regulated state li = 1 or an upregulated state

hi = 1 depend in a Hebbian manner on presynaptic activity and the

state of the postsynaptic neuron. In the absence of presynaptic

activity, the LTD rate rL vanishes. Presynaptic activity combined

with a time-averaged membrane potential u above a critical value

qLTD leads in the TagTriC model to a LTD transition rate rL

proportional to [u(t)2qLTD]. For a transition from the initial state

to the high state, we require in addition that the momentary

membrane potential is above a second threshold qLTP. Hence the

transition rate rH is proportional to [u(t)2qLTD][u2qLTP]

Author Summary

Humans and animals learn by changing the strength ofconnections between neurons, a phenomenon calledsynaptic plasticity. These changes can be induced byrather short stimuli (lasting sometimes only a few seconds)but should then be stable for months or years in order tobe useful for long-term memory. Experimentalists haveshown that synapses undergo a sequence of steps thattransforms the rapid change during the early phase ofsynaptic plasticity into a stable memory trace in the latephase. In this paper we introduce a model with a smallnumber of equations that can describe the phenomena ofinduction of synaptic changes during the early phase ofsynaptic plasticity, the trigger process for protein synthe-sis, and the final stabilization. The model covers a broadrange of experimental phenomena known as taggingexperiments and makes testable predictions. The ability tomodel the stabilization of synapses is crucial to understandlearning and memory processes in animals and humansand a necessary ingredient for any large-scale model of thebrain.

TagTriC-Model of Early and Late LTP/LTD


Figure 1. The three components of the Tag-Trigger-Consolidation (TagTriC) model. (A) A synapse can be in the non-tagged state N, thehigh state H or the low state L. A synapse i in H (or L) has a tag hi = 1 (or li = 1, respectively). Transitions to a tagged state occur with rates rH forpotentiation and rL for depression. The tag hi = 1 is indicated by a red flag in both the flow graph and the schematic drawing below. (B) Synthesis ofplasticity related proteins p (green squares) is triggered if the total number of set tags is larger than a critical number Np. If the trigger threshold Np isnot reached, the protein concentration decays back to zero. (C) The consolidation dynamics can be visualized as downward motion in a potentialsurface E(z). The function f(z) (shown to the right) is the derivative of E and characterizes the dynamics dz/dt = f(z). If a tag is set at the synapse (hi = 1)and protein synthesis has been triggered (p<1), the dynamics can be imagined as downward motion into the right well of the potential E(z). In thiscase, z<1 is the only fixed point of the dynamics (magenta circle). In the absence of tags (hi = li = 0, below) the consolidation variable zi of synapse i isbistable and approaches (direction of flow indicated by arrows) stable fixed points at zi = 0 or zi = 1 (magenta circles). The steps of synaptic taggingand capture are indicated immediately below the flow diagram. (D) The tagging rates for depression (2rL,(magenta)) and for potentiation rH (blue)are shown as a function of the clamped voltage under the assumption that a presynaptic spike has arrived less than 1 millisecond before. Note thatfor depression we plot the negative rate 2rL rather than rL to emphasize the fact that depression leads to a down-scaling of the synapse. (E) Voltagedependence of early LTP and LTD. The weight change Dw/w(0) induced by a stimulation of 100 synapses at 2 Hz during 50 s while the postsynapticvoltage is clamped is shown as a function of voltage. The percent change Dw/w in simulations (circles) of LTP/LTD induction experiments can bepredicted from a theory (solid line) based on the difference in transition rates rH2rL. The simulation reflects the voltage dependence seen inexperiments [5,39]. (F,G) Frequency dependence of early LTP and LTD. Simultaneous stimulation of 100 synapses by 3 trains (separated by 5 min) of100 pulses at rates ranging 0.03 to 100 Hz shows LTD at low frequencies and LTP at frequencies above 30 Hz. (G) If LTP is blocked in the model, LTD(pink line) occurs up to high frequencies as in experiments [7]. Blue line: LTP with blocked of LTD.doi:10.1371/journal.pcbi.1000248.g001



whenever these threshold conditions are satisfied; see Methods for

details.

Our assumptions regarding the transition rates essentially

summarize the qualitative voltage dependence seen in the

Artola-Brocher-Singer experiments [5]. Indeed, when 100 synap-

ses in the TagTriC model are stimulated at low frequency during

50 seconds while the membrane voltage is kept fixed at different

values (Figure 1D), the total weight change summed across all

synapses exhibits LTD at low voltage and LTP at high voltage

[38,39]. As expected, the resulting weight changes in the

simulations of Figure 1E reflect the voltage dependence of the

transition rates in Figure 1D.

Trigger for Protein SynthesisPreviously induced LTP or LTD needs to be consolidated in

order to last for more than one hour. Consolidation requires that

protein synthesis is triggered. Experimental evidence indicates that

triggering of protein synthesis needs the presence of neuromod-

ulators such as dopamine (in the apical CA1 region) or other

modulators (in other regions). In typical tagging experiments,

extracellular stimulation co-stimulates dopaminergic input leading

to a phasic dopamine signal [13,40]. In our model, induction of E-

LTP or E-LTD through appropriate stimulation protocols changes

the synaptic efficacy and sets tags at the modified synapses, both

described by the variables hi = 1 or li = 1. Protein synthesis in the

model is triggered (see methods for details) if the total number of

tags Si(hi+li) (which indirectly reflects the phasic dopamine signal)

reaches a threshold Np which depends on the level of background

dopamine (and other neuromodulators). More specifically, Np

decreases with the concentration of background dopamine so that

the presence of dopamine facilitates the trigger process [32].

If the trigger criterion is satisfied, the concentration p of

synthesized plasticity related proteins approaches with rate kp a

value close to one. If the number of tags falls below the threshold

Np, the protein concentration p decays with a time constant tp back

to zero. Further details on the role of the trigger threshold and its

relation to neuromodulators can be found in the discussion section.

Consolidation and Late LTPThe total weight wi of a synapse i depends on the present value

of the tags hi or li as well as on its long-term value zi. The slow

variable zi is a continuous variable with one or two stable states

described by a generic model of bistable switches, that could be

implemented by suitable auto-catalytic processes [16]. While the

concentration p of plasticity related proteins is zero, the variable zi

has two stable states at zi = 0 and zi = 1, respectively. If the protein

concentration takes a value of p<1, one of the stable states

disappears and, depending on the tag that was set, the long term-

value of the synapse can be up- or down-regulated; see methods

and Figure 1C for details.

In order to illustrate the mechanism of induction of L-LTP, let

us suppose that the synapse has been initially close to the state

zi = 0. The dynamics of the synapse can be imagined as downward

motion in a ‘potential’ E. The current stable state of the synapse is

at the bottom of the left well in the potential pictured in Figure 1C.

We assume that during a subsequent LTP induction protocol the

synapse has been tagged with hi = 1 and that the total number of

tags set during the LTP induction protocol surpasses the trigger

threshold Np. If the protein concentration p approaches one, the

potential surface is tilted so that the synapse now moves towards

the remaining minimum at z<1. After decay of the tags, p returns

to zero, and we are back to the original potential, but now with the

synapse trapped in the state z = 1. It can be maintained in this state

for a long time, until another strong tagging event occurs during

which the synapse is tagged with li = 1 as a result of LTD

induction. In this case the potential surface can be tilted towards

the left so that the only equilibrium point is at z = 0. Since

consolidation is typically studied in animals that are more than 20

days old [13], we assume that before the beginning of the

experiment 30 percent of the synapses are already in the

upregulated state z = 1 and the remaining 70 percent in the state

z = 0; see also [7]. Because of the bistable dynamics of

consolidation, only synapses that are initially in the upregulated

state z = 1 can undergo L-LTD and only synapses that start from

z = 0 can undergo L-LTP; compare [7]. Note, however, that tags

for potentiation and depression can be set independently of the

value of z. We may speculate that the variable z is related to the

activity of PKMf [11,14], or to the self-sustained clustering of

AMPA receptors [41], but the exact biochemical signaling chain is

irrelevant for the functional consequences of the model discussed

in the results section. In our model, the bistable dynamics of the z-

variable captures the essence of synaptic persistence despite

molecular turnover [15,16,28] and mobility of AMPA receptors

[41].

Tests of the ModelThe TagTriC model has been tested on a series of stimulation

protocols that reflect induction of LTP and LTD as well as the

consolidation of plasticity events.

Induction of Synaptic ChangesA typical LTP induction experiment starts with extracellular

stimulation of a bundle of presynaptic fibers (i.e., the Schaffer

collaterals leading from CA3 to CA1) that activate a large number

(typically hundreds [13]) of presynaptic terminals. With an

extracellular probe electrode placed close to one of the

postsynaptic neurons, a change in synaptic efficacy is measured

via the amplitude (or initial slope) of the evoked postsynaptic

potential, representing the total response summed across all the

stimulated synapses. In our simulations, we mimic these

experiments by simultaneous stimulation of 100 synapses. The

state of the postsynaptic neuron is described by the adaptive

exponential integrate-and-fire model [42] and can be manipulated

by current injection.

In a preliminary set of simulation experiments done with

presynaptic stimulation alone (no manipulation of the postsynaptic

neuron), the TagTriC model exhibits LTD or LTP depending on

the frequency of the presynaptic stimulation (Figure 1F) in

agreement with experimental results [4,43]. Moreover, under the

assumption that LTP has been blocked pharmacologically (rH = 0

in the model), our model shows LTD even for high stimulation

frequencies (Figure 1G). This stems from the fact that LTD and

LTP are represented in the TagTriC model by two independent

pathways (Figure 1A) which are under control condition in

competition with each other, but show up individually if one of the

paths is blocked [43]. Together with the voltage dependence of

Figure 1E, the above simulation results indicate that our model of

LTP and LTD induction can account for a range of experiments

on excitatory synapses in the hippocampal CA1 region, in

particular, voltage and frequency dependence.

Consolidation of Synaptic ChangesIn order to study whether consolidation of synaptic changes in

our model follows the time course seen in experiments, we

simulate standard experimental stimulation protocols [12,13]. A

weak tetanus consisting of a stimulation of 100 synapses at 100 Hz

for 0.2 seconds (21 pulses) leads in our model to the induction of

LTP (change by +15 percent) which decays back to baseline over



the time course of two hours (Figure 2A). Thus, after the early

phase of LTP the synapses are not consolidated. A stronger

stimulus consisting of stimulating the same group of hundred

synapses by 100 pulses at 100 Hz (repeated 3 times every

10 minutes) yields stronger LTP that consolidates and remains

elevated (weight change by 2265 percent) for as long as the

simulations are continued (more than 10 hours, only the first

5 hours are shown in Figure 2B). Thus our model exhibits a

transition from early to late LTP if E-LTP is induced by the strong

tetanic stimulation protocol, but not the weak one, consistent with

results in experiments [12,13]. If, however, the weak tetanus at a

first group of 100 synapses is given 30 minutes before or after a

strong tetanus at a second group of 100 synapses, the synapses in

both the weakly and strongly stimulated groups are consolidated

(Figure 2C and 2D). If the weak tetanus in group one is given

120 minutes after the strong tetanus in group two, then

consolidation of the synapses in the weakly stimulated group does

not occur (Figure 2E). Thus our model exhibits a time course of

heterosynaptic interaction between the two groups of synapses as

reported in classical tagging experiments [12,13].

An advantage of a modeling approach is that we can study the

dependence of the heterosynaptic interaction between the two

groups of synapses upon model parameters. A critical parameter in

the model is the trigger threshold Np that needs to be reached in

order to start protein synthesis (Figure 1B). With our standard

choice of parameters, where Np = 40, we can plot the consolidated

weight change Dw/w(0) in the weakly stimulated group (measured

10 hours after the induction) as a function of the time difference

between the stimulation of the group receiving the strong tetanus

and that receiving the weak tetanus. The curve in Figure 2F shows

that for a time difference up to 1 hour there is significant

interaction between the two groups of synapses leading to synaptic

consolidation, whereas for time differences beyond 2 hours this is

no longer the case. If the trigger threshold is increased to Np = 60

(corresponding to less available neuromodulator), then the

maximal time difference that still yields L-LTP in the weakly

stimulated group of synapses is reduced to about 20 minutes

(Figure 2F) whereas a reduction of Np yields an increased time

window of interaction (data not shown). If Np is reduced much

further, the weak tetanus alone will be sufficient to allow a

transition from the early to the late phase of LTP. We speculate

that Np could depend on the age of the animal as well as on the

background level of dopamine or other neuromodulators so as to

enable a tuning of the degree of plasticity (see discussion for

details).

LTD and Cross-TaggingWe consider two experimental protocols known to induce

LTD—a weak low-frequency protocol consisting of 900 pulses at

1 Hz and a strong low-frequency protocol consisting of 900

repetitions at 1 Hz of a short burst of three pulses at 20 Hz. This

strong low-frequency protocol applied to 100 model synapses leads

to a significant level of LTD (reduction of weights to 7064 percent

of initial value) which is consolidated 5 hours later at a level of

8363 percent of initial value. If a group of 100 synapses is

stimulated with the weak low-frequency protocol, an early phase of

LTD is induced that is not consolidated but decays over the time

course of 3 hours (Figure 3A and 3B). However, if the weak low-

frequency stimulation occurs after another group of 100 synapses

had been stimulated by the strong low-frequency protocol, then

the group that has received the weak stimulation shows

consolidated synapses (at 9062 percent 5 hours after stimulus

induction, Figure 3C). Moreover, consolidation of LTD (at 9263

percent 5 hours after stimulus induction) in the group of synapses

receiving the weak low-frequency protocol also occurs if it was

stimulated thirty minutes after the stimulation of a second group of

synapses by a strong tetanus, leading to LTP (Figure 3D). Thus,

the TagTriC model exhibits cross-tagging consistent with

experiments [11,32]. In our model, cross-tagging occurs because

the tags for LTP and LTD (hi and li, respectively) enter in a

symmetric fashion into the trigger criterion for the synthesis of

plasticity-related proteins (see Figure 1 and Methods).

Model Mechanism for Tagging, Cross-Tagging, andConsolidation

In order to elucidate how the model gives rise to the series of

results discussed in the preceding paragraphs, we have analyzed

the evolution of the model variables during and after induction of

LTP (Figure 4). Critical for consolidation is the synthesis of

plasticity related proteins, characterized by the variable p in the

model. Synthesis is only possible while the total number of tagsPNi hizli is above the protein triggering threshold Np. For the

strong tetanic stimulus this criterion is met for about 90 minutes

(shaded region in Figure 4A) leading to high levels of plasticity

related proteins. After 90 minutes the concentration of proteins

starts to decay back to baseline. While the level of proteins is

sufficiently elevated the consolidation variable zi of each tagged

synapse moves towards zi<1 since this is the only stable fixed point

of the dynamics (Figure 1C). This leads to a consolidation time of

about 2 hours, enough to switch a large fraction of synapses into

the up-regulated state z<1 (green line, Figure 4A). Hence the

average weight of the stimulated synapses stabilizes at a value

above baseline, indicating L-LTP (Figure 4A, solid line).

If, in a different experiment, 100 synapses are stimulated by the

weak tetanus, the synthesis of plasticity related proteins is only

possible during a few minutes (Figure 4B, red line), which is not

sufficient to switch tagged synapses from z = 0 into the upregulated

state z<1. Hence the weights (Figure 4B, black line) decay

together with the tags (Figure 4B, magenta line) back to baseline

and the transition from early to late LTP does not occur. The

decay of the weights is controlled by the rate kH at which tags

stochastically return to zero. The evolution of the protein

concentration p and the consolidation variable z after a strong

tetanus that leads to 90 minutes of protein synthesis and a weaker

tetanus that only leads to 40 minutes of protein synthesis has been

illustrated in (Figure 5A).

The total amount of available protein that is synthesized

depends in our model on the time that the total number of tags

stays above the protein triggering threshold Np. Even though

always 100 synapses are stimulated in our model, not all receive

tags in each experiment; moreover because of the competition for

potentiation tags (hi = 1) and depression tags (li = 1) during

induction of plasticity, different synapses can receive different tags

in the same experiment. With our strong tetanus protocol, on

average 70 (out of 100) synapses receive a potentiation tag and 30

a depression tag while with the weak tetanus the numbers are 30

and 10, respectively. For the depression protocols, on average 10

synapses receive a potentiation tag and 90 a depression tag under

strong low-frequency stimulation, and typically zero a potentiation

tag and 40 a depression tag under the weak low-frequency

protocol. These numbers vary from one trial to the next so that

sometimes the protein trigger threshold Np = 40 is reached with the

weak protocols and sometimes not. The important aspect is that

even if the threshold is reached for a short time, the duration of

protein synthesis is not long enough to provide a sufficient protein

concentration p for consolidation of the tagged synapses; see

Figure 4B and Figure 5A.



Since the concentration p of plasticity related proteins is crucial

for the transition from early to late LTP we wondered how a block

of protein synthesis would interfere with the consolidation of

weights in the TagTriC model. Application of a protein synthesis

inhibitor (modeled by setting the rate kp of protein synthesis to

zero) during 1 hour starting thirty minutes before a strong tetanus

is given to a group of 100 synapses that would normally lead to L-

LTP, induced E-LTP but prevented consolidation into L-LTP

(data not shown). However, if the same simulation experiment was

repeated after a second group of synapses had received a strong

tetanic stimulation 35 minutes prior to the application of protein

synthesis blocker, then both groups of synapses showed consoli-

dation of weights (Figure 4D), consistent with experiments [12].

Closer inspection of the lower panel in Figure 4D shows that two

components contribute to consolidation: Firstly, the concentration

of plasticity related proteins (red line) that has increased because of

Figure 2. The model accounts for tagging paradigms. (A) A weak tetanus (21 pulses at 100 Hz) applied at a group of 100 synapses att = 10 min (arrow) leads to an increased connection weight (w/w(0), blue line) that decays back to baseline. (B) A strong tetanus (100 pulses at 100 Hzrepeated three times, arrows) leads to late LTP that is sustained for 5 hours (black line). (C) If the weak tetanus (blue arrow) in a first group of synapsesis followed thirty minutes later by a strong tetanus (black arrows) in a second group of synapses, the weights in the first group (blue line) and thesecond group (black line) are stabilized above baseline. (D) Stimulating a group of synapses by a weak tetanus (blue arrow) 30 minutes after the endof the strong tetanic stimulation of a second group also leads to stabilization of the weights in both groups above baseline. (E) If the weak tetanicstimulation occurs 2 hours after the strong tetanic stimulation of the other group, only synapses in the strongly stimulated group will be stabilized(black line), but not those in the weakly stimulated group (blue line). (F) Fraction of stabilized weights Dw/w(0) in the weakly stimulated groupmeasured 10 hours after induction of LTP as a function of the time difference between the weak stimulation and the end of the strong tetanicstimulation in the second group. Blue line: normal set of parameters (Np = 40). Black line: protein trigger threshold increased to Np = 60. In panels A–E,lines indicate the result averaged over 10 repetitions of the simulation experiments and bars standard deviation. In panel F, line indicates the resultaveraged over 100 repetitions. 90 of the 100 individual trials stayed within the bounds indicated by the error bars.doi:10.1371/journal.pcbi.1000248.g002



the first strong tetanic stimulus decreases only slowly back to

baseline enabling the switching of the slow components (variable z,

green line) even in the presence of protein synthesis blocker.

Secondly, even after the end of the application of the blocker, the

total number of tags that has been set by LTP induction is still

above the critical value Np (shaded region in Figure 4D) so that

protein synthesis can be resumed after the end of the blocking

period. In summary, the detailed analysis of the TagTriC model

allows to account for many aspects of tagging experiment in terms

of a limited number of variables.

Discussion

Relation of Models to ExperimentsSynaptic plasticity is based on intricate signal transduction

chains involving numerous processing steps and a large number of

different molecules [2,13,17]. Despite the complexity of the

molecular processes, synaptic plasticity has experimentally been

characterized by a small set of distinct phenomena such as short-

term plasticity [44] as well as early and late phases of LTP and

LTD [13].

Existing models of synaptic plasticity have focused on the

description of short-term plasticity [44] and on the induction of

LTP and LTD [24–26,33–36]. The question of maintenance has

received much less attention and was mainly addressed in the

context of bistability of the CaMKII auto-phosphorylation process

[27–29], AMPA receptor aggregation [41], or four identified

kinase pathways [45]. While CaMKII is necessary for induction of

long-term potentiation [46], it is probably too narrow to focus

modeling studies only on a single or a few kinases such as CaMKII

and neglect other proteins and signaling cascades that are involved

in synaptic maintenance [13]. For example, there is strong

evidence that PKMf is involved in synaptic maintenance and

necessary for the late phase of LTP in vitro [11] and in vivo [14].

However, the actual processes are complex and the molecules

involved in setting tags may differ between different parts of the

dendrite. For example PKMf is involved in setting tags during E-

LTP in the basal dendrite, whereas CaMKII (or MAPK for E-

LTD) plays a similar role in apical dendrites [30].

Instead of focusing on specific signaling cascades, the TagTriC

model presented in this papers aims at describing the essential

ingredients of any possible functional model of L-LTP and

tagging. These ingredients include (i) a bistable switch (described

by the dynamics of the zi-variable) for each synapse that

guarantees long-term stability in the presence of molecular turn-

over [16]; (ii) a global triggering signal for protein synthesis

(described by the dynamics of the p variable); a formalism to (iii)

induce early forms of LTP and LTD and (iv) set synaptic tags.

Since we aimed for the simplest possible model, we have identified

the synaptic tags hi and li for potentiation and depression with the

Figure 3. The model accounts for cross-tagging between LTP and LTD. (A) A strong low-frequency stimulus (3 pulses at 20 Hz, repeated 900times every second) applied to a group of N = 100 synapses induces LTD with mean weights (w/w(0)) stabilized at 8363% of initial value after 5 hours(black line). (B) A weak low-frequency stimulus (1 pulse repeated 900 times at 1 Hz) induces early LTD, which is not consolidated. (C) If the weak low-frequency stimulus is applied 30 minutes after a second group of synapses has received the strong low-frequency protocol, the weights in bothgroups (blue, weak stimulus; black, strong stimulus) are consolidated at values below baseline. (D) Consolidation of LTD in the group receiving weaklow-frequency stimulation (blue line) also happens if induction occurs 30 minutes after stimulating a second group of synapses with a strong tetanicprotocol (see Figure 2) inducing LTP (black line). Downward arrows indicated the period of weak (blue arrow) or strong (black arrow) low-frequencyprotocols. The black upward arrows indicate strong tetanic stimulation. Lines show mean results, averaged over 10 repetitions of the simulationexperiment. Error bars are standard deviation.doi:10.1371/journal.pcbi.1000248.g003



synaptic weights during the early phase of LTP and LTD,

respectively, so that points (iii) and (iv) are described by the same

transition of the synapse from an initial non-tagged state to the

high or low state, respectively. Variants of the model where the

weight during the early phase of LTP and LTD is not directly

proportional to the value of the tags are conceivable.

Even though we do not want to identify the synaptic variables hi,

li, zi with specific biochemical signals, a couple of candidate

molecules and signaling chains should be mentioned. The setting

of the tag for LTP under normal physiological conditions involves

NMDA receptor activation and elevated levels of calcium which in

turn trigger a signaling chain involving Calmodulin and CaMKII.

We therefore think that the hi variable (representing both the tag

for LTP induction and the weight increase during the early phase

of LTP) should be related to the activation of CaMKII [13,46].

The molecular interpretation of the tag li for LTD is less clear

[13]. In our model we have taken the tags as discrete quantities

that decay stochastically, but a model with continuous tags that

decrease exponentially gives qualitatively the same results (data not

shown). The reason is that triggering protein synthesis in our

model requires a large number of tags to be set, so that even in the

stochastic model only the mean number of tags is relevant–and the

mean (more precisely, its expectation value) is a continuous

variable. Nevertheless, we prefer the model with discrete values

over the continuous one in view of the switch-like transitions of

synapses after induction of LTP and LTD [7,37]. Maintenance of

enhanced synaptic weights is probably implemented by an

increased number of AMPA receptors in the postsynaptic

membrane. Whether the stability arises from a self-organization

process of receptors [41] or from interaction with persistently

activated CaMKII molecules [46] or from additional kinases such

as PKMf [11,14], is an open problem of experimental

investigation. Similarly, the exact identity of many plasticity

related proteins is still unknown [13]. In our model we assume that

recently synthesized plasticity related proteins are accessible to all

synapses onto the same postsynaptic neuron. However, a

distinction between proteins synthesized in, say, basal dendrites

and that synthesized in apical dendrites would be possible by

Figure 4. Dynamics of the TagTriC Model during different tagging protocols and protein synthesis blocking. The change of the totalsynaptic weight (top panels, black line Dw~

PNi~1 wi tð Þ{wi 0ð Þ=N½ �) has contribution from early LTP (top panels, magenta line representsPN

i~1 hi{ali=Nð Þ) and from late LTP (top panels, green line representsPN

i~1 b zi{zi 0ð Þð Þ=N). The protein variable p (red line, bottom panels) growsas long as the average number of tags (

PNi~1 hizlið Þ=N , blue line) is above the protein synthesis trigger threshold (Np/N, dashed horizontal line). For

better visibility, the regions where the blue line is above the trigger threshold is shaded. (A) A strong tetanus (N = 100 synapses, stimulated by 100pulses at 100 Hz, repeated three times every ten minutes) leads to a sustained period of about 90 minutes where the number of tagged synapses isabove the protein synthesis triggering threshold (lower panel, blue shaded). During this time the protein synthesis variable p is close to one (red line,lower panel), causing an increase in the fraction of consolidated weights (green line, top panel). (B) During a weak tetanus (N = 100 synapses,stimulated by 21 pulses at 100 Hz) the number of tags surpasses the protein triggering threshold only for a short time which does not enableswitching of the z variable (top panel, green line) to the up-regulated state. (C) If the weak tetanus is given 30 minutes after the strong one, thenumber of tags set by the strong tetanus is still above the threshold, which allows protein synthesis stabilizing both the group of 100 synapsesreceiving the strong tetanus (top panel) and the group of 100 synapses receiving the weak tetanus (middle panel). (D) Protein synthesis is blocked for1 hour (indicated by black bar at bottom of panel) starting 35 minutes after a first group of 100 synapses has been stimulated by a strong tetanus.Despite protein synthesis blocking, both the first group of synapses (top panel) and a second group of 100 synapses that received a strong tetanusduring the blocking period (middle panel) develop late LTP because proteins synthesized during the induction of early LTP in the first group decayonly slowly (bottom panel).doi:10.1371/journal.pcbi.1000248.g004



replacing the variable p by two or more distinct variables pk with

similar dynamics (but potentially different trigger thresholds Np),

allowing for a compartmentalization of tagging [13].

Experimental cross-tagging results clearly indicate that there are

two different types of synaptic tags, one for LTP and one for LTD

[13,32], which we called hi for LTP and li for LTD, leading to

three different states during tagging (Figure 1A). Since we have

identified the tagging with the early phase of LTP and LTD, our

model of E-LTP and E-LTD also has three different states

(whereas our model of late LTP/LTD has only two states

characterized by zi = 0 and z2 = 1). The three-state model of early

LTP/LTD presented in this paper would predict that all non-

tagged synapses can undergo a transition to E-LTP or E-LTD

depending on the induction protocol–whereas experiments suggest

that about 70 percent of synapses show LTP but not LTD and the

remaining 30 percent LTD but not LTP [7]. Moreover, only those

synapses that are initially weak can be potentiated and only those

that are initially strong can be depressed [7]. This aspect can be

included in our model if we replace the induction rates rH for LTP

by rH(12zi) and rL for LTD by rlzi so LTP is only possible from a

state with zi = 0 and LTD only from an initial state zi = 1 — in

agreement with a two-state model of early LTP/LTD [7]. For the

tagging and induction experiments presented in this paper, the

results do not change significantly when we implement this

extension of the induction model.

Functional Consequences and PredictionsOne of the advantages of a simple phenomenological model is

that it should be capable of illustrating the functional consequences

of tagging and L-LTP or L-LTD in a transparent manner. What

are these functional consequences?

A characteristic feature that is made transparent in our model

(and which we expect to be present in any model of tagging) is

that, under typical experimental conditions, the transition from

early to late LTP is only possible if a sizable group of synapses have

undergone E-LTP or E-LTD. Hence, while induction of E-LTP is

a local Hebbian process that is likely to take place at the

postsynaptic site of the synapse (e.g., the dendritic spine), the

transition from the early to the late phase of LTP requires a

minimum number of synapses to be activated by appropriate

stimulation including co-activation of neuromodulatory input so as

to trigger synthesis of plasticity related proteins. A direct

consequence of this is that synapses cannot be considered as

independent. In order to predict whether a synapse memorizes an

item for a long time or forgets it and re-learns some other item, it is

not sufficient to consider a ‘Hebbian’ induction model, where

synaptic changes depend only on the activity of pre- and

postsynaptic neurons. For maintenance, it is not the synapse

which decides individually, but it is the neuron as a whole (or a

large functional compartment sharing the same site of synthesis of

plasticity-related proteins [13,30,47]) which ‘decides’ whether it is

going to store the present information, or not. Hence, classical

[20,21,34] and recent [22] theoretical models which studied

memory maintenance in the presence of ongoing neuronal activity

on the level of single synapses need to be reconsidered, since the

assumption of independent synapses does not hold (Figure 5A and

5B). In particular, our model predicts that, after an ensemble of

identical neurons have received the same stimulus, some neurons

learn (adapt a large fraction of their synapses to the stimulus) and

others don’t (keep all their synapses unchanged). With our choice

of parameters, this happens in the TagTriC model if the number

of synapses that have been tagged during the induction protocol is

between 55 and 70 (Figure 5B). This neuronal, rather than

synaptic, decision about memorizing an input (see also [48]) is

potentially attractive for prototype learning–a standard paradigm

in neuronal clustering and categorization algorithms, e.g., [19]. In

contrast to traditional neuronal clustering models where learned

Figure 5. Theory and predictions. (A) Evolution of the variables p and z during tagging. If protein synthesis is ‘ON’ and the synapse tagged, p andz move along the black dashed line towards the stable fixed point on the upper right (p<1, z<1) (red filled circle). If protein synthesis stops aftersome time (yellow line, after 90 min; orange line, after 40 minutes) but the synapse remains tagged, the dynamics converges towards the fixed pointp = 0, z = 1 (red filled circle) indicating that the synapse is consolidated (yellow and orange trajectories). However, if protein synthesis stops too early(after 25 min, pink line), or if the synaptic tag is lost too early (after 60 min, magenta line), the synapse is not consolidated and the trajectoriesconverge towards the non-tagged initial state p = 0, z = 0 (red filled circle). The green dashed vertical line at z = 0.5 indicates the threshold beyondwhich a loss of the tag does not affect consolidation; the green solid line indicates the separatrix between the stable fixed points at z = 0 and z = 1.The minimal duration of protein synthesis to allow any consolidation is given by the intersection of the black dashed line with the separatrix. (B)Number of consolidated synapses (Nup, vertical axis) as a function of the number of initially tagged synapses (Ntag, horizontal axis) in simulations (redfilled circles) and theory (solid line). Some of the initially tagged synapses fail to be consolidated because either they lose their tag or proteinsynthesis stops too early (see A). With a protein synthesis threshold Np = 40 (arrow) we need about 60 initially tagged synapses to achieve anyconsolidation (solid line). If the protein synthesis threshold is reduced to Np = 10 (dashed arrow), we need at least 15 tagged synapses to see anyconsolidation (dashed line).doi:10.1371/journal.pcbi.1000248.g005



memories need to be protected against overwriting by completely

different memory items [19], a model based on tagging would

have an intrinsic vigilance threshold via the trigger threshold Np.

Hence it is resistant to changes at a single synapse.

In our view, the protein synthesis trigger threshold NP is an

important control parameter in the model. The results of Figure 2F

show that an increase of the trigger threshold reduces the maximal

delay after which a weak tetanus leads to L-LTP after a strong

tetanic stimulation in a different group of synapses. With our

normal value of Np = 40 we need around 60 synapses to be initially

tagged in order to retain any memory. If we decrease the trigger

threshold to Np = 10 and keep all other parameters of the model

unchanged, then we need at least a group of 15 synapses tagged

during the induction protocol to get any consolidation since some

of the initially tagged synapses loose their tag too early to get

consolidated (Figure 5B). Only for a very small trigger threshold,

say Np = 1, (which could occur at high concentration of

neuromodulators) synapses become (nearly) independent, since a

tag at a single synapse would be sufficient to trigger the synthesis of

proteins which would then become available at that synapse.

Repeated stimulation of the synapse alone would then be sufficient

to transform E-LTP into L-LTP.

In our opinion, the trigger threshold Np is significantly lower in

the presence of neuromodulators such as, for example, dopamine

(for synapses from Schaffer collaterals onto CA1 pyramidal

neurons) or noradrenaline (for synapses in the dentate gyrus). A

simple model for the dependence of Np on dopamine would be

Np = n0/(DAbg+c0) where n0 is some arbitrary number (say n0 = 1),

c0 a small number (say 0.001) and DA denotes the stationary

‘background’ concentration of dopamine (that is, before the start

of the experiment), normalized to 0,DAbg,1. The phasic

dopamine signal caused by co-stimulation of dopaminergic input

during tagging experiments is assumed to be proportional to the

number of tagsPN

i hizli. The trigger conditionPN

i hizliwNp

becomes then equivalent to the conditionPNi hizli

� �DAbgzc0

� �wn0 which shows a trade-off between

the phasic dopamine signal and the stationary background level of

dopamine. In particular in the presence of a large concentration of

dopamine (DA<1), single synapses can be consolidated. With the

assumption that standard tagging experiments in a large group of

synapses are performed at a low dopamine concentration of

DA = 0.024 before stimulation, we retrieve the value of Np = 40

used in the main part of the results section. The dependence of the

trigger criterion on the number of tagsPN

i hizli takes implicitly

the co-activation of neuromodulatory input during the experi-

mental stimulation protocol into account: the larger the number of

stimulated neurons and the stronger the stimulus, the higher the

probability of co-activation of dopaminergic fibers. Blocking

dopamine receptors amounts in the model to setting both the

background and the phasic dopamine signal to zero. In this case,

protein synthesis is not possible.

Our model of LTP/LTD induction does not only account for

voltage and frequency dependence of LTP/LTD induction, but

also for spike timing dependence. In fact, for a stimulation

paradigm where postsynaptic spikes are induced by short current

pulses of large amplitude either a few milliseconds before or after

presynaptic spike arrival, the model of LTP/LTD induction used

in the TagTriC model becomes formally equivalent to a recent

model of spike-timing dependent plasticity [35] which can be seen

as an extension of classical models of STDP [24–26]. In the case of

stochastic spiking of pre- and postsynaptic neurons our model

shares important features with the Bienenstock-Cooper-Munro

model [33], in particular the quadratic dependence upon the

postsynaptic variables. In addition, our model also accounts for the

voltage dependence of the Artola-Brocher-Singer model [38].

Thus, the model of LTP/LTD induction shares features with

numerous established theoretical models and covers a large range

of experimental paradigms known to induce LTP or LTD [3–6,8].

Since the subsequent steps of protein synthesis trigger and

stabilization are independent of the way early phase of LTP is

induced, our model predicts that tagging experiments repeated

with different stimulation paradigms, but otherwise identical

experimental preparation and age of animal, should give similar

results as standard tagging protocols. In particular we propose to

stimulate a group of synapses in hippocampal slices by 40–60

extracellular current pulses at 10 Hz while the postsynaptic

neuron is receiving intracellular current injection that triggers

action potential firing either a few milliseconds before or after

presynaptic spike arrival and keeps the membrane potential at a

depolarized level between postsynaptic action potential firing. Our

model predicts that this will induce early LTD or LTP depending

on spike timing and depolarization level that is not maintained

beyond 1 or 2 hours. However, if the same stimulation occurs after

a second group of synapses has received a strong tetanus, then

stabilization of synapses at potentiated or depressed levels should

occur, similar to standard tagging and cross-tagging experiments.

In our opinion, these predictions should not depend on model

details, but hold for a broad class of models that combine a

mathematical description of induction of synaptic plasticity with a

mechanism of consolidation.

Another finding—which is somewhat unexpected and in

contrast to other conceptual models of synaptic tagging and

capture [12,13,47]—is that during a strong tetanic stimulation a

fraction of synapses receives tags for depression (while most, but

not all, receive tags for potentiation). This is due to the fact that

during induction of plasticity, transition to E-LTP and E-LTD act

in parallel [7]. The prediction is that after consolidation (say

2 hours after the strong tetanic stimulation) a small fraction of

synapses would show L-LTD, rather than L-LTP.

An essential ingredient of our model that allows long-term

stability of consolidated synapses is the bistable dynamics of the

variable z. In our opinion, such bistability (or possibly multi-

stability [49] with three or four stable states) is necessary for

synaptic maintenance in the presence of molecular turn-over, as

recognized in earlier theoretical work [15,16,34]. Our model

therefore predicts that L-LTP and L-LTD should have bistable,

switch-like properties. While there is evidence for switch like

transitions during the induction of E-LTP and E-LTD [7,37], the

bistability of the late phase of synaptic plasticity has so far not been

shown. A possible experiment would be to combine a minimal

stimulation protocol (e.g., a weak tetanus) at a single synapse

[7,37] with a medium to strong stimulus at a group of other

synapses (e.g., tetanic stimulus varying between 30 and 100 pulses).

The prediction is that the weight of the single synapse shows an all-

or-none phenomenon with transition probabilities that depend on

the stimulation of the group of other synapses. In particular, as the

number of pulses of the tetanic stimulation is reduced (covering a

continuum from strong to weak tetanic stimulation), the

maintenance in the potentiated state should become less likely

(averages across many experiments decrease) whereas the results of

individual experiments show either full potentiation or none,

which should give rise to a bimodal distribution of normalized

synaptic weights.

Open Questions and PerspectivesA lot of questions remain open and need to be addressed in

future studies. First, can a synapse that has been potentiated in the

past and is maintained after a transition to late LTP undergo a



further potentiation step [13]? In our current model this is not

possible since the consolidation variable z has only two stable fixed

points. If we replace the function f(z) depicted in Figure 1 by

another one with more than two stable fixed points, then the

answer to the above question would be positive. Indeed, there

have been suggestions that self-organization of receptors into

stable sub-groups could lead to multiple stable states [49].

Second, induction of LTP or LTD is not only possible by

strong extracellular stimulation of groups of synapses, but also at

single synapses if presynaptic activity is paired with either a

depolarization of the postsynaptic membrane [5,7] or tightly

timed postsynaptic spikes as in STDP experiments [6,8]. How

can it be that the change induced by STDP seems to be

maintained over one hour without visible degradation? [6,7].

Are synapses in these experiments consolidated, and if so what is

the concentration of neuromodulators? In the TagTriC model

with the choice of parameters used in the present paper,

consolidation would not be possible, since the minimum number

of synapses that have undergone E-LTP or LTD is Np = 40 in

order to trigger protein synthesis, but, as explained above, an

increased neuromodulator concentration would make consolida-

tion possible.

Third, what is the role of NMDA receptor activation during

synaptic consolidation? In our present model, protein synthesis is

triggered by appropriate induction protocols, but is independent of

synaptic activity during the consolidation process. However, recent

experimental results suggest that protein synthesis blocker needs

synaptic stimulation during the consolidation period to become

effective [50], suggesting a subtle interplay between protein

synthesis and synaptic activation that cannot be captured by our

model.

Fourth, has each neuron a single protein synthesis unit or is

protein synthesis a local process confined to each dendritic

branch? In the first case, there is a single neuron-wide protein

synthesis trigger threshold [12] and the neuron as a whole

‘decides’ whether early forms of synaptic potentiation and

depression will be consolidated or not. This is the paradigm

posited in the TagTriC model. In the alternative model of local

protein synthesis [13,47], the critical unit for consolidation are

local groups of synapses on the same dendritic branch. Thus, for

the same number of tagged synapses, a local group of synapses

on the same dendritic branch is more likely to undergo

consolidation than a distributed set of tagged synapses, leading

to a form of clustered plasticity [47]. The TagTriC model can

be easily adapted to the case of clustered plasticity by (i)

replacing the point-neuron model by a neuron model with

spatially distributed synapses and (ii) replacing the neuron-wide

trigger equation (see 4 and Figure 1B) by a finite number of

analogous, but dendrite-specific equations.

Fifth, how can tags be reset? Experiments show that a

depotentiating stimulus given 5 minutes after a weak tetanus

erases the trace of E-LTP (resets the tag) whereas depotentiation

10 or 15 minutes after the strong tetanus only transiently

suppresses the E-LTP, making the consolidation of the synapse

by protein capture possible [51]. We have checked in additional

simulations that our present model cannot account for these

experiments. In our opinion, the above tag-reset experiments show

that the synapse has additional hidden states currently not

included in the TagTriC model. Additional states would allow

to (i) separate the measured early LTP during the first 5 minutes

from setting the tag; and (ii) distinguish between depotentiation

and depression of synapses. One interpretation of the tag-reset

experiments [51] is that during the first five minutes the tag is not

yet set whereas early LTP is already visible. The tag would be set

only with a delay of 5–10 minutes. Application of a depotentiating

stimulus more than 10 minutes later would then leave the

potentiation tag intact, but move the synapse to a transiently

depotentiated state.

The final and potentially most interesting question is that of

functional relevance: Can the TagTriC model be used to simulate

reward-based learning in experiments in vivo [13]? The formal

theory of reinforcement learning makes use of an eligibility trace

[52] which can be interpreted as a synapse specific tag. In the

future we want to check whether the TagTriC model can be linked

to reinforcement learning models [53–56] under the assumption

that reward prediction errors are represented by a dopamine

signal [57] which influences the protein synthesis dynamics in our

model. This open link to reward-based learning is of fundamental

functional importance.

Methods

Model of Early LTP/LTD and TaggingIn our model we assume that presynaptic spike arrival needs to

be combined with a depolarization of the postsynaptic membrane

(e.g., [5]) in order to induce a change of the synapse. In voltage

clamp experiments (e.g., [39]) the postsynaptic voltage would be

constant. However, in general the voltage is time-dependent and

described by a variable u(t). In the TagTriC model, we assume that

the low-pass-filtered voltage

u tð Þ~ 1

tlowP

ð?0

exp {s

tlowP

� �u t{s{eð Þds:

needs to be above a critical value qLTD to make a change of the

synapse possible. tlowP is the time constant of the low-pass filter

and e = 1 ms is a short delay twice the width of a spike (see

Table 1). This short delay ensures that u includes effects of

previous presynaptic inputs and postsynaptic spikes, but not of an

ongoing postsynaptic action potential.

Table 1. Parameter values used throughout all simulations,except Figure 1E–G where Np = 10 and initial percentage ofzi = 1 was 10%, because these simulations refer toexperiments with younger animals.

Tag Trigger Consolidation

N = 100 kp = 1/(6 min) N = 100

ALTD = 0.01 tp = 60 min c = 0.1

ALTP = 0.014 Np = 40 tz = 6 min

tx = 100 ms b = 2

tLTPlowP~100 ms Initialisation:

N(zi = 1) = 30

tLTDlowP~1 s

e= 1 ms

kh = 1/h

kl = 1/(1.5 h)

HLTD = 270.6 mV

HLTP = 250 mV

a = 0.5

Initialisation: li = hi = 0

doi:10.1371/journal.pcbi.1000248.t001



Combining presynaptic spike arrival at synapse i (represented by

xi) with a depolarization u of the postsynaptic neuron above a

threshold qLTD we get a rate of LTD

rL~ALTDxi tð Þ u tð Þ{qLTD½ �z ð1Þ

where ALTD.0 is a parameter and [.]+ denotes rectification, i.e.,

[y]+ = y if y.0 and zero otherwise. Here xi tð Þ~P

f d t{tfi

� �denotes the presynaptic spike train with pulses at time t

fi and d the

Dirac-delta function. Formally, rL describes the rate of stochastic

transitions from the non-tagged state h = 0, l = 0 to the low state

l = 1, Figure 1. In simulations we work with discrete time steps of

D= 1 ms. Eq. 1 indicates that the probability Pl = 0Rl = 1 of a

transition to the low-state during the time step D vanishes in the

absence of presynaptic spike arrival and takes a value of

Pl = 0Rl = 1 = 12exp(2ALTD[u(t)2qLTD]+D)<ALTD[u(t)2qLTD]+D if

a presynaptic spike arrives at the synapse i during the time step D.

Note that the transition from l = 0 to l = 1 is only possible if h = 0

and h remains zero during the transition.

Similarly, a switch from the non-tagged state h = 0, l = 0 to the

high state h = 1 occurs at a rate rH which also depends on

postsynaptic voltage and presynaptic spike arrival. We assume that

each presynaptic spike at synapse i leaves a trace xi that decays

exponentially with time constant tx. The exact biophysical nature

of the trace is irrelevant, but could, for example, represent the

amount of glutamate bound to the postsynaptic receptor. The

value of the trace at time t caused by earlier spike arrivals at time

tfi is then xi tð Þ~ 1=txð Þ

Pf exp { t{t

fi

� �.tx

h iwhere the sum

runs over all firing times tfi vt. With the trace xi we write

rH~ALTPxi tð Þ u tð Þ{qLTD½ �z u tð Þ{qLTP½ �z ð2Þ

which indicates that, in addition to the conditions for LTD

induction we also require the momentary membrane potential u(t) to

be above a second threshold qLTP. This threshold could change on

the time scale of minutes or hours as a function of homeostatic

processes. To summarize, the rate of LTP transition rH is different

from rL in five aspects. First, the constant ALTP is not the same as

ALTD. Second, LTP is caused by the trace xi left by presynaptic

spikes, rather than the spikes themselves. This trace-formulation

ensures that presynaptic spikes can interact with later postsynaptic

spikes as in classical models of STDP [24–26]. Third, the time

constant of the low-pass filter in u is different; fourth, the

momentary voltage needs to be above a threshold qLTP; and fifth,

the total dependence upon the postsynaptic voltage is quadratic,

rather than linear. The quadratic dependence ensures that for

large depolarization LTP dominates over LTD [39]. Tagged

synapses with hi = 1 decay with probability Ph = 1Rh = 0 = kHD back

to the non-tagged state (and analogously, but with rate kL for the

transition li = 1Rli = 0).

In the TagTriC model, the local synaptic values h = 1 for

potentiation or l = 1 for depression act as tags indicating potential

sites for further consolidation, but are also directly proportional to

the weight of the synapse after induction of LTP or LTD. Since in

minimal stimulation experiments LTD leads to a reduction of

about 50 percent of the synaptic efficacy whereas LTP leads to an

increase by up to 100 percent [7], we model the weight change

during the early phase of LTP as Dwi = (hi2ali)w where w is the

weight of the non-tagged synapse and a = 0.5. The total weight

change Dw/w measured shortly after induction of LTP or LTD

with extracellular protocols corresponds to the fraction of synapses

in the high or low states, respectively, hence, if all synapses start

from the non-tagged state the measured weight change is

Dw.

w~PN

i~1 hi{alið Þ=N~ShT{aSlT where N is the number

of synapses stimulated by the protocol. The set of parameters of

LTP/LTD induction and tagging is given in table 1.

TriggerThe triggering process is controlled by the dynamics of a variable

p which describes the amount of plasticity related proteins

synthesized in the postsynaptic neuron. Protein synthesis is triggered

and the variable p increases while the concentration of dopamine

exceeds a critical level qp [58]. If the dopamine concentration DA

falls below qp, the protein concentration decays with a time constant

tp. Assuming standard first-order kinetics we have

dp

dt~kp 1{pð ÞH DA{qp

� �{

p

tp

ð3Þ

Protein synthesis has a maximum rate dp/dt of kp and saturates if the

amount of protein approaches a value one. H[y] denotes the unit

step function with H[y] = 1 for y.0 and zero otherwise.

Dopamine is present at a low stationary background value. In

addition a phasic dopamine component is induced in standard

tagging experiments in hippocampal slices, because of co-

stimulation of dopaminergic inputs during extracellular stimula-

tion of presynaptic fibers [40]. To describe the time course of the

phasic dopamine component in our model, we assume that the

dopamine is proportional to the total number of tags Si(hi+li)

induced by the stimulation protocol. The stationary background

level of dopamine DAbg is included in the threshold qp = Np(DAbg)

for protein synthesis. Hence Eq. 3 can be rewritten in the form

dp

dt~kp 1{pð ÞH

Xi

hizlið Þ{Np DAbg

� �" #{

p

tp

ð4Þ

Note that we have chosen units so that the threshold for protein

synthesis Np can be interpreted as the minimal number of tags

necessary to stimulate protein synthesis. This interpretation is

important for the discussion of the model results, in particular

Figures 4 and 5.

A suitable model for dependence of the protein synthesis

threshold on the background level of dopamine is Np(DAbg) = n0/

(DAbg+c0) where n0 = 1 is a scaling factor, c0 = 0.001 a constant and

0#DAbg#1 is the normalized dopamine concentration. We note

that the trigger condition [Si(hi+li)2Np(DAbg)].0 is then equiva-

lent to the condition (DAbg+0.001)[Si(hi+li)].1. This formulation

shows that there is a trade-off between background levels and

phasic dopamine. Unless stated otherwise we always use in the

simulation a fixed dopamine level DAbg = 0.024 so that Np = 40.

The specific model Np(DAbg) of the dependence upon background

dopamine levels is therefore irrelevant.

We assume that the plasticity related protein p synthesized in the

postsynaptic neuron is diffused in the dendrite of the postsynaptic

neuron and hence available to all the synapses under consider-

ation. Hence, the tags hi and li have indices, since they are synapse-

specific, whereas p in Eq. 4 does not.

Consolidation and Late LTPThe consolidation variable z describes the late phase of LTP

and follows the dynamics

tz

dzi

dt~f zið Þzc DAð Þ hi{lið Þp: ð5Þ



The scaling factor c is a function of the dopamine level DA. In the

simulations we always assumed a fixed dopamine level and set

c(DA) = 0.1.

In the absence of plasticity related proteins (p = 0), or if no tags

are set (hi = li = 0), the function f(z) = z(12z)(z20.5) generates a

bistable dynamics with stable fixed points at z = 0 and z = 1 and an

unstable fixed point at z = 0.5 marked by the zero crossings of the

function f, Figure 1C. In the presence of a finite amount of

proteins p.0 and a non-zero tag, the location of the fixed points

changes and for p.0.47, only one of the stable fixed points

remains. The potential shown in Figure 1C is a function E with

dE/dz = 2f(z) so that dz/dt = 2dE/dz. We note that a synapse i can

change its consolidated value only if both a tag (hi = 1 or li = 1) and

protein p.0.47 is present–summarizing the essence of ‘synaptic

tagging and capture’ [12,13].

Synaptic WeightThe synaptic weights have contributions from early and late

LTP and LTD. The total synaptic weight of a synapse i is

wi = w(1+hi2ali+bzi) where w is the value of a non-tagged synapse,

a = 0.5 and b = 2 are parameters, hi and li are binary values

indicating E-LTP and E-LTD, respectively, and zi is the value of

the L-LTP trace of synapse i. Since we model slice experiments in

animals older than 20 days, we assume that 30 percent of the

synapses have undergone previous potentiation and have z = 1

while the remaining 70 percent of synapses are in the state z = 0

[7]. In all simulation experiments we stimulate one or several

groups of N = 100 synapses each. Assuming that no tags have

been set in the recent past (hi = li = 0), the initial value of the

average weight in a group of N synapses is then

w 0ð Þ~wPN

i~1 1zbzi

h i.N~1:6w.

Neuron ModelFor all simulations in this paper we use the adaptive exponential

integrate-and-fire model [42] as a compact description of neuronal

firing dynamics. Briefly, it consists of two equations. The voltage

equation has an exponential and a linear term as measured in

experiments [59]. The second equation describes adaptation.

Although firing rate adaptation is not important for the present

study, it would be relevant in the context of other stimulation

paradigms. Parameters for the neuron model are as in [42] and are

kept fixed for all simulations presented in this paper. The voltage

threshold Vs of spike initiation by a short current pulse is 25 mV

above the resting potential of 270.6 mV [42]. Synaptic input is

simulated as a short current pulse. The initial connection weight w

was adjusted so that simultaneous activation of 40 or more

synapses triggers spike firing in the postsynaptic neuron. Hence the

amplitude of a single EPSP is about 0.6 mV.

The adaptive exponential integrate-and-fire model is defined in

continuous time. If a spike is triggered by a strong current pulse,

the voltage rises within less than 0.5 millisecond to a value of

20 mV where integration is stopped. The voltage is then reset to

resting level, and integration restarted after a refractory time of

1 ms. In order to enable us to perform simulations of plasticity

experiments with a time step of D= 1 ms, the voltage equation

during the rising slope of the action potential was integrated once

at a much higher resolution (time step 0.02 ms), so as to determine

the exact contribution of each postsynaptic spike to the probability

of LTP induction. Every postsynaptic spike was then treated as an

event in the plasticity simulations that contributed a probability

Ph = 0Rh = 1 of flipping the tag from h = 0 to h = 1 in a time step

D= 1 ms which we can write as Ph = 0Rh = 1 = aDx(t)[u(t)2qLTD]+

with a numerical conversion factor aD = ALTP 5 ms mV derived by

the above procedure; see Eq. 2.

Number of Consolidated SynapsesIn Figure 5 we plot the number of synapses that have been

consolidated as a function of the number Ntag of initially tagged

(hi = 1) synapses. Since the number of tags decays exponentially

with rate kH, the expected duration TONP of protein synthesis is

TONP ~ 1=kHð Þln Ntag

Np

� �where Np is the protein trigger

threshold. While protein synthesis is ‘ON’ the variables p and z

move along the black dashed line in Figure 5A which crosses after

a time t1 the separatrix (green line in Figure 5A) and at a time t2the line z = 0.5 (vertical dashed green line). Different cases have to

be distinguished. (i) TONP vt1, no consolidation takes place (see

pink trajectory), hence Nup = 0. (ii) TONP wt2, consolidation is

guaranteed for all synapses that are still tagged at time t2, hence

Nup = Ntagexp(2kt2). (iii) In the case of t1vTONP ƒt2, the time tcross

needed to cross the vertical line z = 0.5 is numerically calculated by

integrating the equations dp/dt = 2p/(tp) and dz/dt = f(z)+c p

starting at t~TONP at the point p TON

P

� �,z TON

P

� �on the black-

dashed line (see orange line in Figure 5A for a sample trajectory).

The number of consolidated synapses is then Nup = Ntagexp(2ktcross).

The solid line in Figure 5B represents Nup as a function of Ntag

calculated for the cases (i)–(iii). With our standard set of parameters,

we have t1<28 min and t2<60 min.

Acknowledgments

We thank Julietta Frey and Mark van Rossum for helpful discussions.

Author Contributions

Conceived and designed the experiments: CC WG. Performed the

experiments: CC. Analyzed the data: CC. Wrote the paper: WG. Designed

the model of late LTP and performed research: LZ. Participated in

research on a precursor model of early LTP/LTD: EV LB.

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Tag-Trigger-Consolidation: A Model of Early and Late Long ... · protein kinase Mf (PKMf) [11,14]. The stabilization and maintenance of potentiated synapses poses a number of theoretical